For those who already know what Platonic Solids are, one might wonder as to whether there are any other 5 Platonic Solids than the ones which we already know?

For those who do not know what Platonic Solids are, they are those solid objects whose sides are made of a unique regular polygon i.e. they are regular polyhedra.

The 5 platonic solids are:

Now the question arises. Why do we have only 5 Platonic Solids? Are there any other regular polyhedra which haven’t been discovered yet?

In mathematics we always make a logical analysis to find an equation to solve a particular type of problem with available data or to find a solution to a puzzle.

In the above case the puzzle is the existence of only 5 regular polyhedra. To solve this puzzle, first we shall look at the available data and build a formula to verify the existence of a given regular polyhedra with a given number of sides with a given regular polygon.

The data available is that for a given polyhedra there is only one regular polygon which makes up its sides. The other data which is required to form any solid object regular or irregular (i.e. with irregular and/or different polygons ) is that the sum of the angles between the sides of the polygons at any given vertex of the polyhedra should always be less than 360 degrees else the polyhedra will flatten out at that particular vertex and the formation of a closed solid object would be impossible.

So we have two sets of data:

• Sum of the angles of all the sides at any given vertex  of the regular polyhedra has to be less than 360^o
• The number of sides which meet at a vertex is same for all vertices (for a regular polyhedra) and the angles between all sides and the total angle at any vertex is the same at all vertices (as all the sides are made up of regular polygons).

Thus if ‘n’ is the total number of the sides which meet at a vertex of the polyhedra and ‘Ø’ is the angle between any two sides of the regular polygon, then the sum of  the angles between all the sides at any vertex of the polyhedra is given by

Ø+Ø+Ø+Ø+Ø+Ø.. n times = nØ

We know that for the polyhedra to exist, nØ hs to be less than 360

i.e. nØ < 360    – (1)

Now we shall see how many regular polyhedra are possible which satisfy the above condition.

We know that a minimum of 3 sides are required to meet at a vertex to form a polyhedra. Thus n>2 is also an essential condition.

• Therefore, let us start with n=3, i.e. let 3 regular polygons meet at each vertex. Let the first regular polygon we start with be the one with the smallest number of sides i.e. a equilateral triangle. So we have Ø = 60. Thus nØ = 3 x 60 = 180 < 360. Hence a regular polyhedra with the above combination is possible as this satisfies our condition (1) and we know that this a tetrahedron.
• Now let n=4 and we continue with an equilateral triangle as the regular polygon. We have
  nØ = 4 x 60 = 240 < 360. Since our condition is satisfied, we have another regular polygon i.e. an Octahedron.
• Let n=5 and we still continue with an equilateral triangle as our regular polygon and we have nØ = 5 x 60 = 300 which still satisfies our condition and here we have a Icosahedron.
• Let n= 6 and we continue with the equilateral triangle i.e. Ø=60. Thus
  nØ = 6 x 60 = 360 = 360. Here our condition (1) is not satisfied and hence a regular polyhedra with this combination cannot exist in nature. With Ø=60 no other regular polyhedra is possible for n =6 or greater.
• So now we switch to the next smallest regular polygon i.e. a square. Hence we have Ø=90. As usual, we shall start with n=3. We have Ø=90 for a square. Thus nØ = 3 x 90 = 270 < 360. Thus it is possible to have a regular polyhedra made up of squares with 3 squares meeting at each vertex. This is our most familiar Platonic Solid i.e. a cube.
• Now let n=4 and Ø=90 i.e. let us continue with the square. Now nØ = 4 x 90 = 360 = 360. Our condition (1) is not satisfied. Hence a regular polyhedra with this combination is not possible nor is it possible for higher values of n with Ø = 90.
• Let us now switch to the next polygon i.e. a pentagon which has 5 equal sides and 5 interior angles with each angle being 108 degrees i.e. Ø=108. Starting with n=3 we have
  nØ = 3 x 108 = 324 < 360. Hence such a regular polyhedra can exist and it is a Dodecagon.
• Now when n=4 and Ø=108 we have nØ = 4 x 108 = 432 > 360. Hence this regular polyhedra cannot exist as our condition (1) is not satisfied. Similarly it cannot exist for higher values of n with Ø=108.
• Now we move on to hexagon as our regular polygon i.e. Ø=120. starting with n=3 we see that
  nØ = 3 x 120 = 360 = 360 . Our condition (1) is not satisfied and this regular polyhedra cannot exist nor can it exist for any higher values of n or Ø as all such combinations will be greater than 360.

Thus we have only 5 Platonic Solids whose existence we have proved in the above reasoning and we also have proved that no other regular polyhedra can exist in nature.

Subscribe to HitXP: Be the first to read.